Generalized Distributive Law

Question

This is problem 3.11, p.29, Introduction to Set Theory, Hrbacek and Jech.

$$ \big( \bigcap_{a \in A} F_a \big) \cup \big( \bigcap_{b \in B} G_b \big) = \bigcap_{(a,b) \in A \times B} (F_a \cup G_b). $$

It is my attempt.

$ \begin{equation} x \in (LHS) \\ \Leftrightarrow x \in \bigcap_{a \in A} F_a ~~ or ~~ x \in \bigcap_{b \in B} G_b \\ \Leftrightarrow (x \in F_a ~~for ~~all~~a\in A)~~or~~ (x\in G_b~~ for~~all~~b\in B). \end{equation} $

And,

$ \begin{equation} x\in(RHS) \\ \Leftrightarrow x \in F_a \cup G_b~~for ~~all~~(a,b) \in A \times B \\ \Leftrightarrow (x\in F_a~~or ~~ x \in G_b) ~~for ~~all~~(a,b) \in A \times B. \end{equation} $

I want to change the position of phrases in each last sentences, but I'm not sure doing it preserves if and only if condition.

Please help my problem.

Answer 1

It is more handsome to do the second part by proving that: $$x\notin\text{ LHS }\implies x\notin\text{ RHS }$$

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Doing it your way the following statements are equivalent:

• $\forall a\in A\;\forall b\in B\;[x\in F_a\vee x\in G_b]$
• $\forall a\in A\;[x\in F_a\vee\forall b\in B\;[x\in G_b]]$
• $\forall b\in B\;[x\in G_b]\vee\forall a\in A\;[x\in F_a]$

This on base of the rule: $$\forall x\;[ Q\vee P(x)]\iff Q\vee\forall x\;P(x)$$